reserve i,j,n,k,m for Nat,
     a,b,x,y,z for object,
     F,G for FinSequence-yielding FinSequence,
     f,g,p,q for FinSequence,
     X,Y for set,
     D for non empty set;
reserve
  B,A,M for BinOp of D,
  F,G for D* -valued FinSequence,
  f for FinSequence of D,
  d,d1,d2 for Element of D;
reserve
  F,G for non-empty non empty FinSequence of D*,
  f for non empty FinSequence of D;
reserve f,g for FinSequence of D,
        a,b,c for set,
        F,F1,F2 for finite set;

theorem Th96:
  for E be Enumeration of F st len E = n + 1 holds
    E|n is Enumeration of F\{E.len E} &
    <*E.len E*> is Enumeration of {E.len E} &
    F = (F\{E.len E}) \/ {E.len E}
proof
  let E be Enumeration of F such that
A1: len E = n + 1;
  set L=len E;
A2: rng <*E.L*> = {E.L} by FINSEQ_1:38;
A3: E = (E|n)^<*E.L*> by A1,FINSEQ_3:55;
  then
A4: E|n is one-to-one & <*E.L*> is one-to-one by FINSEQ_3:91;
A5: F = rng E = rng (E|n) \/ rng <*E.L*> & rng (E|n) misses rng <*E.L*>
    by A3,FINSEQ_1:31,FINSEQ_3:91,RLAFFIN3:def 1;
  then rng (E|n) = F \ {E.L} by A2,XBOOLE_1:88;
  hence thesis by A5,FINSEQ_1:38,A4,RLAFFIN3:def 1;
end;
